but we know geometrically,in complex plane to compute i^-i we take e^(ipi/2)*(-i) and to compute -i^i we take e^(-ipi/2)*i

The exponents ipi/2 and -ipi/2 are complex numbers as such. So multiplying them by i or - i means rotating them in opposite directions.

If we take i*(pi/2) and multiply by -i, we rotate it to the right, clockwise
If we take -i*( pi/2) and multiply it by i, we rotate it to the left, anticlockwise.

in both cases we end up with pi/2 but where did we lose the information about the direction of rotation we had to apply taking selfroots of i and -i ? The directions were opposite, but we end up with 1 number. I do not like it at all, this loss of information.

Now we know that taking self root of i, or -i is like backtetrating from i or -i to e^pi/2. This must happen in some steps via some space, like infinite number series or limits travel through real number space .

Why we do not see that space where these opposite turns of i and -i are happening with every backtetration? Do we miss some dimension? Like spin dimension in physics?